A vast amount of numerical methods have been developed to solve KdV-type equations, including finite difference method, finite volume method, finite element method as well as spectral method. In this paper, we are interested in further exploring discontinuous Galerkin (DG) method, which is a class of finite element method using discontinuous piecewise polynomials for numerical solutions [1]}, [2]}. The main advantages of DG method include flexibility in handling geometry, provable convergence properties and accommodating \(h\) -\(p\)  adaptivity. Various DG schemes have been applied to solve the KdV-type equations. In [3]}, [4]}, [5]}, local DG (LDG) schemes are investigated to solve the KdV-type equations, including the ZK equation. Hybridizable DG (HDG) is used to solve 1D KdV equations in [6]}. In contrast of LDG schemes which need to introduce auxiliary variables and rewrite the equation into first order system, direct DG (DDG) schemes [7]} and ultra-weak DG (UWDG) schemes [8]}, [9]}, [10]} are developed to simulate the 1D KdV equation. While optimal \(L^2\) -error estimates of various types of semi-discrete DG schemes are obtained for the 1D KdV equation [11]}, [6]}, [10]}, [9]} using various projection techniques, only sub-optimal error estimates of LDG schemes are available for the ZK equation [5]}. Also, the ZK equation, which contains mixed derivative term, has not been considered yet under the UWDG framework. One aim of this work is to design stable and accurate UWDG scheme for the ZK equation, which can also shed light on the design of the UWDG formulation for general mixed derivative terms.
We showed that by applying integration by parts in a proper order and continuing to integrate on the trace of elements until the technique can no longer be used, the resulting weak formulation will yield a \(L^2\)  stable scheme.
Further, a novel local projection can be designed to prove optimal \(L^2\)  error estimate on regular meshes.
